What are the Tensions in a System of Fixed Point Charges?

[scott85213]

A fixed point charge of +2q is connected by strings to point charges of +q and +4q, as shown below. Find the tensions T1 and T2. (Use the following as necessary: q, d and k.)

For T1, I summed all the forces on each charge and got
[URL]http://img1.uploadscreenshot.com/images/orig/9/25122291919-orig.png[/URL]
And fairly similar for T2
I have a feeling I'm relatively close, I just need a push in the right direction. Do I need an "I Hat" for the final answer?

 

[collinsmark]

Hello Scott85213,

Welcome to Physics Forums!

A fixed point charge of +2q is connected by strings to point charges of +q and +4q, as shown below. Find the tensions T1 and T2. (Use the following as necessary: q, d and k.)

For T1, I summed all the forces on each charge and got
[URL]http://img1.uploadscreenshot.com/images/orig/9/25122291919-orig.png[/URL]
And fairly similar for T2
I have a feeling I'm relatively close, I just need a push in the right direction.

Close, but not quite.

We can assume that the middle charged is fixed in place: "A fixed point charge of +2q is connected by..." Not that that really matters much, but it will make my following explanation more clear.

Since we are assuming that the middle charge is fixed in place, we can assume that one end of the T₁ string is fixed in place. So the only thing that determines the tension of T₁ is the force on the leftmost charge (the one with a +q charge). Forces on any of the other charges have no resulting effect on the tension T₁.

So I understand where you got the 2q²k/d² term. That is the force resulting from the middle charge (the one that has charge 2q).

I also understand where you got the 4q²k/(2d)² term. That is the force resulting from the rightmost charge (the one that has charge 4q).

But I don't understand where the 8q²k/d² term comes from. That term appears to be the force on the rightmost charge from the center charge. But that force (which happens to be equal and opposite between those two charges) has no bearing on T₁.

Think of it another way. Let's assume that everything is in equilibrium. Now grab on to the 4q charge (with a rubber glove) and pull it to the right with all your might. Assuming the string does not stretch (assume it is very sturdy) and the charge is unchanged (rubber glove), how does your pulling affect T₁? (Remember, the center charge is fixed in place).

Do I need an "I Hat" for the final answer?

I wouldn't. Tension is assumed to be along the direction of the string, wire, line, etc. already. And because tension inherently implies equal and opposite forces on each end of the string, a unit vector does't make much sense. Tension is sort of a scalar quantity. You only need to tack on a unit vector if you need to consider the force from one end of the string in isolation. But for this problem, my final answer would just be the magnitude.

 

[scott85213]

Thanks, the fixed made it much easier. I guess I just read the question too quickly.